When selecting an oscilloscope for a specific
measurement, the first thing that most of us consider is the
bandwidth we’ll need to accurately reconstruct our
signals. After all, the scope’s bandwidth tells us what
spectral frequencies will be preserved and the maximum signal
transition speeds that can be accommodated.

Oscilloscopes are designated by their nominal
bandwidth, such as “the 500-MHz Model XYZ” and
most even have the bandwidth specification embedded in their model
number,

However, this “banner”
specification only describes the maximum bandwidth allowed by the
scope’s front-end circuitry. A scope’s
effective bandwidththe maximum frequency components of a signal
you’ll be able capture, store, and displayis determined
by its sampling rate, which, in turn, can be constrained by the
depth of its acquisition memory.

Briefly exploring the relationship among
bandwidth, sampling rate, and memory depth can provide an
understanding of the tradeoffs involved in selecting a scope and
how to mitigate their effect so as to make measurements with more
confidence.

A quick visit with Dr. Nyquist

The familiar Nyquist-Shannon sampling theorem
states that a signal can be reconstructed exactly if:

the signal is band limited and the sampling frequency is greater
than twice the signal bandwidth.

If we can assume that all samples are equally
spaced in time, then any oscilloscope must maintain a sampling rate
of twice its nominal bandwidth to avoid bandwidth degradation in
the captured signal.

However, this theorem also assumes a theoretical
filter, called a “brickwall” filter, that not
only passes all frequency components below the
bandwidth’s top frequency limit, but also eliminates all
frequency components above this bandwidth (see Fig. 1). A high-performance oscilloscope
with hardware/software brickwall filtering may be able to
accommodate a sampling rate as low as 2.5 times bandwidth. But for
mainstream oscilloscopes, such filters are generally impractical,
and undesirable.

In a typical mainstream oscilloscope, the filter
rolloff is not as aggressive (see Fig.
2). These filters can be implemented more economically, and
their time-domain response is more predictable. The tradeoff is
that you must employ a more conservative sampling rate,
oversampling the bandwidth by a multiple of 4x.

As long as we maintain this 4x oversampling, the
scope’s nominal bandwidth is maintained. However,
anything that causes a reduction in sampling rate will lead to
aliasing below the nominal bandwidth frequency.

Memory’s role

Memory and sampling rate are intertwined
specifications. Because scopes have a fixed display window at any
particular time-per-division (t/div) setting, there are few
settings where both time and memory are maximized. However, it is
more important to maintain the data acquisition (sampling) rateand
therefore the bandwidth of the scopethan it is to use all the
memory.

A simple calculation can tell you how many data
points are required to fill your display: pts per waveform =
sampling rate x t/div x number of divisions.

Consider, for example an oscilloscope with a
5-Gsample/s sampling rate and 10 time divisions set to 100 ns/div.
Then the number of points per waveform is equal to 5 x 109 pts/s x 100 x 109 s/div x 10 div, or 500 points.

As long as the oscilloscope has enough memory to
fill the display, the sampling rate can be sustained. However, if
the sampling rate is so high it would result in data exceeding the
maximum amount of memory, the sampling rate must be reduced to fill
the allotted time.

How sampling rate is reduced with slower sweep
speeds is easily grasped graphically (see Fig. 3). For two hypothetical
500-MHz-bandwidth oscilloscopes, the oscilloscope with more memory
can sustain a high sampling rate over more settings. So why does
that matter? Let us return to our Nyquist analysis.

Scope 1 will oversample the maximum bandwidth by
a factor of 8 at all t/div settings above 500 ns/div (see Fig. 3a), at which point the sampling rate
begins to drop. However, it is not until the sampling rate drops
below 2 Gsamples/s (4x oversampled) that aliasing becomes a
concern. This occurs at 1 μs/div. At that point, any
decrease in sampling rate causes the scope’s effective
bandwidth to drop (see Fig. 3b).

Implications

The above analysis leads us to three
conclusions:

Bandwidth is constrained by the effective sampling rate of the
oscilloscope.Sampling rate can be degraded at slower t/div (sweep)
speeds.Increasing acquisition memory can delay the onset of
sampling rate degradation.

How does this impact your scope selection and
debugging methodology? Well, it really depends upon the signals you
are viewing.

If you spend most of your time with simple
signals like rising edges and transient events, it’s easy
to match your scope’s timebase to the spectral content of
your waveforms – fast edges require fast sweep
speeds.

If you look at more complex signals that combine
slow events and fast events (like modulated signals or trending
signals), you should consider replacing a shallow memory
oscilloscope (fewer than 100 ksamples) with a deeper memory model
(at least 1 Msample).

If you cannot change your current equipment, you
may want to break your analysis into manageable steps. Use slower
t/div settings to characterize slower trends; then switch to faster
settings to characterize high-bandwidth-signal events. If you
choose this path, you may want to use the above calculations to
plot the t/divbandwidth relationship of your scope.

For single-shot acquisitions, the tradeoff
between bandwidth and effective sampling rate is identical, but the
mental model and implications are slightly different. In a
single-shot acquisition, you want to sample for as long as you can
(as long as your measurement requires) as fast as you can. A high
sampling rate is important for maintaining signal fidelity while
zooming in on a signal trend for details about individual
transitions. It allows accurate measurements of both macro- and
micro-events in a one acquisition. If you can’t maintain
a high sampling rate (bandwidth), these events should be measured
with separate acquisitions.

A final word

While the information presented here should help
in understanding important attributes of scope performance, bear in
mind that it has been a somewhat cursory examination of the
relationship between bandwidth, sampling rate, and memory. The
subject of bandwidth is much more nuanced, with factors like
passband flatness and frequency rolloff deserving much more
attention than can be given in the scope of a brief article.

To explore this topic in greater depth, two
application notes are of particular value: Evaluating Oscilloscope Sample Rates vs. Sampling
Fidelity: How to Make the Most Accurate Digital Measurements
(AN-1587) and Choosing an
Oscilloscope with the Right Bandwidth for Your Application
(AN-1588). Both of these application notes can be found online at
http://www.agilent.com using
the site’s search function